Triangle A B C has centroid G. Lines are drawn from each point to the midpoint of the opposite side to form line segments A D, B E, C F.

Question

Triangle A B C has centroid G. Lines are drawn from each point to the midpoint of the opposite side to form line segments A D, B E, C F.
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In triangle ABC, BG = 24 mm. What is the length of segment GE?

12 mm
24 mm
36 mm
48 mm

GPT-4o mini · Answer

In triangle $ ABC $, the centroid $ G $ divides each median in a 2:1 ratio. Since $ D $, $ E $, and $ F $ are the midpoints of sides $ BC $, $ CA $, and $ AB $ respectively, the segments $ AD $, $ BE $, and $ CF $ are the medians. Given that $ BG = 24 $ mm, and knowing that $ G $ is located 2/3 of the way along median $ BE $, this means that segment $ GE $ is the remaining 1/3 of the median. Since $ BG $ is 24 mm (the portion from vertex $ B $ to the centroid), segment $ GE $ will be: $$ GE = \frac{1}{2} \times BG = \frac{1}{2} \times 24 \text{ mm} = 12 \text{ mm} $$ Thus, the length of segment $ GE $ is: $$ \boxed{12 \text{ mm}} $$